A New Fast Numerical Method for One-Dimensional Nonlinear Sine-Gordon Equation Using Multivariate Pad\'e approximation

TL;DR

This paper introduces a novel fast numerical method based on multivariate Padé approximation for solving the 1D nonlinear Sine-Gordon equation, demonstrating high accuracy and ease of application for multidimensional problems.

Contribution

It presents a new definition of multivariate Padé approximation and applies it to efficiently solve the nonlinear Sine-Gordon equation.

Findings

Numerical results match analytical solutions closely.

The method is computationally efficient.

Applicable to multidimensional problems.

Abstract

This paper has two purposes. First we present a new definition of the multivariate Pad\'e approximation, a new fast numerical method. Then numerical solution of the one-dimensional (1D) time-dependent nonlinear Sine-Gordon equation (SGE) is considered by this method. Numerical results are obtained for various cases involving undamped SGE. The results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme. It is shown that the technique is easy to apply for multidimensional problems.